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How to automatically generate short and easy-to-understand proofs for geometric theorems has long been an issue of concern in mathematics education. A novel automated geometric theorem proving method based on complex number identities is proposed in this paper, which acts as a bridge between geometry and algebra. According to the proposed method, the geometric relations in the given proposition are first transformed into a complex number expression, then the complex number identity is generated by the elimination method; finally, the closure property under all four operations of real numbers is employed to prove the proposition. A test on more than 300 geometric problems shows that the proposed method is highly effective, and the corresponding proofs are short, with obvious geometric meaning. [ABSTRACT FROM AUTHOR]

This article introduces the space of A-linearly correlated fuzzy complex numbers. Using this space, we study the stationary Schrödinger equation with boundary conditions are given by fuzzy complex numbers. This equation plays an special role in Quantum Mechanics describing the state of the system. We apply the formalism to the step potential, generating quantum results consistent with traditional quantum results. [ABSTRACT FROM AUTHOR]

The fixed-time synchronization (FTS) of complex-valued memristive competitive neural networks (CVMCNNs) with mixed delays is the main topic of this paper. Firstly, two new fixed-time (FT) stability criteria are obtained. In the FT stability criterion, the exponent is usually positive, but the exponent can be less than zero and varies with the error state in this paper. Then, based on two new stability theorems, the FTS of CVMCNNs is studied in the sense of 1-norm and 2-norm, respectively, without dividing the complex-valued state variables into the real part (RP) and the imaginary part (IP). Finally, two numerical simulation examples are given to further demonstrate the validity and superiority of our results. [ABSTRACT FROM AUTHOR]

Given a bounded operator Q on a Hilbert space H , a pair of bounded operators (T 1 , T 2) on H is said to be Q-commuting if one of the following holds: T 1 T 2 = Q T 2 T 1 or T 1 T 2 = T 2 Q T 1 or T 1 T 2 = T 2 T 1 Q. We give an explicit construction of isometric dilations for pairs of Q-commuting contractions for unitary Q, which generalizes the isometric dilation of Ando (Acta Sci Math (Szeged) 24:88–90, 1963) for pairs of commuting contractions. In particular, for Q = q I H , where q is a complex number of modulus 1, this gives, as a corollary, an explicit construction of isometric dilations for pairs of q-commuting contractions, which are well studied. There is an extended notion of q-commutativity for general tuples of operators and it is known that isometric dilation does not hold, in general, for an n-tuple of q-commuting contractions, where n ≥ 3 . Generalizing the class of commuting contractions considered by Brehmer (Acta Sci Math (Szeged) 22:106–111, 1961), we construct a class of n-tuples of q-commuting contractions and find isometric dilations explicitly for the class. [ABSTRACT FROM AUTHOR]

Erdös–Lax inequality relates the sup norm of the derivative of a polynomial along the unit circle to that of the polynomial itself (on the unit circle). This paper aims to extend the classical Erdös–Lax inequality to the polar derivative of a polynomial by using the extreme coefficients of the given polynomial. The obtained results not only enrich the realm of Erdös–Lax-type inequalities but also offer a promising avenue for diverse applications where these inequalities play a pivotal role. To illustrate the practical significance of our results, we present a numerical example. It vividly demonstrates that our bounds are considerably sharper than the existing ones in the extensive literature on this captivating subject. [ABSTRACT FROM AUTHOR]

A complex intuitionistic fuzzy (CIF) set contains the membership and non-membership in the shape of a complex number whose amplitude term and phase term are covered in the unit interval. Moreover, Hamacher interaction aggregation operators are the combination of two major operators, called Hamacher aggregation operators and interaction aggregation operators, and they are used to aggregate the collection of information into one value. In this manuscript, we present the concept of Hamacher interaction operational laws for CIF sets (CIFSs). Further, we develop the CIF Hamacher interaction weighted averaging (CIFHIWA) operator, CIF Hamacher interaction ordered weighted averaging (CIFHIOWA) operator, CIF Hamacher interaction weighted geometric (CIFHIWG) operator, and CIF Hamacher interaction ordered weighted geometric (CIFHIOWG) operator. For these operators, we also discuss some basic properties, such as idempotency, monotonicity, and boundedness. Additionally, we develop a MADM method based on the developed operators and apply it to solve the green supply chain management problems, which can implement environmentally friendly practices to minimize carbon emissions, resource consumption, and waste generation while promoting long-term sustainability. Finally, we verify the superiority and effectiveness of the proposed method based on a comparative analysis between the proposed techniques and existing methods. [ABSTRACT FROM AUTHOR]

We derive computable formulas for the structured backward errors of a complex number λ when considered as an approximate eigenvalue of rational matrix functions that carry a symmetry structure. We consider symmetric, skew-symmetric, Hermitian, skew-Hermitian, ∗ -palindromic, T-even, T-odd, ∗ -even, and ∗ -odd structures. Numerical experiments show that the backward errors with respect to structure-preserving and arbitrary perturbations are significantly different. [ABSTRACT FROM AUTHOR]

Recent progress has revealed a number of constraints that cosmological correlators and the closely related field-theoretic wavefunction must obey as a consequence of unitarity, locality, causality and the choice of initial state. When combined with symmetries, namely homogeneity, isotropy and scale invariance, these constraints enable one to compute large classes of simple observables, an approach known as (boostless) cosmological bootstrap. Here we show that it is possible to relax the restriction of scale invariance, if one retains a discrete scaling subgroup. We find an infinite class of solutions to the weaker bootstrap constraints and show that they reproduce and extend resonant non-Gaussianity, which arises in well-motivated models such as axion monodromy inflation. We find no evidence of the new non-Gaussian shapes in the Planck data. Intriguingly, our results can be re-interpreted as a deformation of the scale-invariant case to include a complex order of the total energy pole, or more evocatively interactions with a complex number of derivatives. We also discuss for the first time IR-divergent resonant contributions and highlight an inconsequential inconsistency in the previous literature. [ABSTRACT FROM AUTHOR]

We show that the construction of the higher Gaudin Hamiltonians associated with the Lie algebra gl n admits an interpolation to any complex number n. We do this using the Deligne’s category D t , which is a formal way to define the category of finite-dimensional representations of the group G L n , when n is not necessarily a natural number. We also obtain interpolations to any complex number n of the no-monodromy conditions on a space of differential operators of order n, which are considered to be a modern form of the Bethe ansatz equations. We prove that the relations in the algebra of higher Gaudin Hamiltonians for complex n are generated by our interpolations of the no-monodromy conditions. Our constructions allow us to define what it means for a pseudo-differential operator to have no monodromy. Motivated by the Bethe ansatz conjecture for the Gaudin model associated with the Lie superalgebra gl n | n ′ , we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic behavior of a class of delayed quaternion-valued neural networks (QVNNs) with impulses and parameter uncertainties. First, we assume that the activation function and connection matrices in the model are defined in the quaternion domain. Without decomposing the QVNNs into four equivalent neural networks in a real number domain or two equivalent neural networks in a complex number domain, the existence and uniqueness of the equilibrium point (EP) are studied based on the M-matrix and homeomorphism mapping theories. By combing the vector Lyapunov function with mathematical induction, theorems are given to ensure robust exponential stability of the system's EP. The results reflect the influence of connection matrices, delays, impulses, and the activation function on the convergence speed of the EP. Finally, the feasibility of the derived results is explained through three examples. [ABSTRACT FROM AUTHOR]

Let ℓ be a prime number different from the residue characteristic of a non-archimedean local field F. We give formulations of ℓ -adic local Langlands correspondences for connected reductive algebraic groups over F, which we conjecture to be independent of a choice of an isomorphism between the ℓ -adic coefficient field and the complex number field. [ABSTRACT FROM AUTHOR]

Let be a polynomial of degree having all its zeros in , , then for every real or complex number with , Govil and McTume [7] showed that the following inequality holds In this paper, we have obtained a generalization of this inequality involving sequence of operators known as polar derivatives. In addition, the problem for the limiting case is also considered. [ABSTRACT FROM AUTHOR]

The subject of this paper is the study of some properties of -Stirling numbers of the second kind for a complex or a -adic complex number. In the -adic setting, as we known, the Laplace transform plays an important role in the study of some arithmetic sequences. We remind the definition of the Laplace transform of a -adic measure and its link with the moment of this measure. With the aid of a specific measure we establish some identities and congruences for the -Stirling numbers when is a non zero -adic complex number and for the generalized -Stirling numbers of the second kind attached to a -adic function that is invariant by . Also, we express the generalized -Stirling numbers according to generalized Stirling numbers . [ABSTRACT FROM AUTHOR]

Based on N different coherent states with equal weights and phase-space rotation symmetry, we introduce N-headed incoherent superposition states (NHICSSs) and N-headed coherent superposition states (NHCSSs). These N coherent states are associated with N-order roots of the same complex number. We study and compare properties of NHICSSs and NHCSSs, including average photon number, Mandel Q parameter, quadrature squeezing, Fock matrix elements and Wigner function. The main characteristic of this study is to use the same complex number to express the related states that appear in a large number of literatures. [ABSTRACT FROM AUTHOR]

Let H be the Radford Hopf algebra of dimension twelve over a field k . In this paper, we investigate the automorphism groups of the Green ring r(H) and the Green algebra ℂ(H) ≔ ℂ ⊗ℤr(H) over the complex number field ℂ. The ring automorphisms of r(H) and the algebra automorphisms of ℂ(H) are all described. It is shown that the ring automorphism group of r(H) is isomorphic to the Klein group K4, and the algebra automorphism group of ℂ(H) is isomorphic to the direct product ℂx × S4, where ℂx is the multiplicative group of all nonzero complex numbers and S4 is the symmetric group of degree 4. [ABSTRACT FROM AUTHOR]

An elliptic curve E can be immersed in P N - 1 as a curve of degree N by means of the linear system of |NO|, where O is the origin of E. Well-known classical results going back to Bianchi and Klein say that if N is odd, this immersion is uniquely determined by specifying a full-level N structure. In this paper we show that if N is even, uniqueness of immersion is ensured by specifying a level structure associated with a certain congruence subgroup between Γ (N) and Γ (2 N) . Moreover, we construct, over the complex number field, an immersion by means of suitably chosen theta functions, and write down the quadratic equations satisfied by them. [ABSTRACT FROM AUTHOR]

The method of linear theory for small perturbations is used to study the stability of heavy inclusions in the upper crust. The existence of such inclusions of increased density is associated with chemical heterogeneity or phase transitions. At small deformations considered in the linear stability theory, the material undergoes transient creep which is described by the Andrade rheological model. With this rheology, the convective instability of thickened segments of the lithosphere is oscillatory: the increment is a complex number with a real part close to zero (threshold instability). Heavy inclusions in the upper crust introduce a small complex correction to the increment that determines the convective instability of the lithosphere, slightly increasing the real and imaginary parts of the increment, enhancing the instability and increasing the oscillation frequency. [ABSTRACT FROM AUTHOR]

Biquaternion algebra is an algebraic structure originating from a complex number and has mainly been used in quantum mechanics, special and general relativity, classical, relativistic, and covariant electrodynamics, and signal processing. In this paper, the problem of the diagonalization of a biquaternion matrix is studied, by means of a complex representation of a biquaternion matrix, and an algebraic algorithm for the diagonalization of a biquaternion matrix is presented. In addition, numerical examples demonstrate the effectiveness of the algebraic algorithm. [ABSTRACT FROM AUTHOR]

Assume that is a faithful normal semifinite trace on a von Neumann algebra , is the unit of , is the -algebra of -measurable operators, and is the Banach space of -integrable operators. We present a new proof of the following generalization of Putnam's theorem (1951): No positive self-commutator with is invertible in . If is infinite then no positive self-commutator with can be of the form , where is a nonzero complex number and is a -compact operator. Given with we seek for the conditions that . If and with then . If and then . If a partial isometry lies in and for some integer then is a commutator and implies that . [ABSTRACT FROM AUTHOR]

In this paper, we consider the spectral problem for the semi-integer Bessel equation with a boundary condition containing the square of the spectral parameter and a complex physical parameter. The system of eigenfunctions of the problem and the characteristic equation for the eigenvalues are derived. The equation for multiple roots of the characteristic equation is derived. The results on the basis properties (Riesz basis) of the system of eigenfunctions at different values of the parameter are obtained. For each case a biorthogonally conjugate system is constructed. At the end of the paper there is an example for the order of Bessel functions equal to . [ABSTRACT FROM AUTHOR]

Inspired by a problem found in a textbook of Faddeev and Sominsky we propose a new form of roots of an arbitrary complex number. Our idea uses the zeros of some polynomials connected with the modified Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study the weighted composition operators W f , φ on the Hardy space H 2 that are (T u , λ) -Toeplitz, where u is a nonconstant inner function and λ is a complex number with | λ | ≤ 1 ; that is, T u ∗ W f , φ T u = λ W f , φ . In particular, we prove that for an analytic self-map φ of D , not identically zero on D , and f ∈ H ∞ \ { 0 } , if λ ∈ D ¯ \ { 0 } such that 1 λ (u ∘ φ) is a self-map of D , then W f , φ is (T u , λ) -Toeplitz precisely when C φ u = λ u . We also give several applications of this result. [ABSTRACT FROM AUTHOR]

In this paper, we develop the notion of c-almost periodicity for functions defined on vertical strips in the complex plane. As a generalization of Bohr's concept of almost periodicity, we study the main properties of this class of functions which was recently introduced for the case of one real variable. In fact, we extend some important results of this theory which were already demonstrated for some particular cases. In particular, given a non-null complex number c, we prove that the family of vertical translates of a prefixed c-almost periodic function defined in a vertical strip U is relatively compact on any vertical substrip of U, which leads to proving that every c-almost periodic function is also almost periodic and, in fact, c m -almost periodic for each integer number m. [ABSTRACT FROM AUTHOR]

This paper covers the modeling and nonlinear control of the mechanical systems where instead of angle coordinates, unit complex numbers are used as control states. This approach is useful not only to get rid of trigonometric functions, but mainly because it is a specific case of the 3D configuration that utilizes unit hypercomplex numbers (quaternions) as system states, and therefore facilitates its understanding, which represents a contribution to control education. As an example of application, it is considered a particular case of a cube balanced on one of its edges, which is equivalent to a reaction wheel inverted pendulum. The derived nonlinear control law is equivalent to a linear one and is characterized by only three straightforward tuning parameters. Experimental results are presented to validate modeling and control. [ABSTRACT FROM AUTHOR]

Complex numbers are basic. An inconsistency would question Wigner's unreasonable effectiveness of mathematics. A vehicle to study this question is Kirchoff's scalar diffraction theory. In the paper, an inconsistency in the real phase angle of a complex number is presented. When this inconsistency is introduced in Kirchoff's theory, we can study its influence on the experimental success of this theory. There are no a priori reasons to include or exclude real phase angles. Referring to Wigner's idea of the role of mathematics in empirical science, an experiment can provide more insight. In the experiment, a weak intensity, small wavelength source can be employed. When the contradictory phase angle is excluded, a nonzero diffraction amplitude appears physically possible. If it is included, this amplitude vanishes. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space and B s (H) the Jordan algebra of all bounded self-adjoint linear operators on H. Assume that k is any positive integer with k < dim H and ξ is any complex number. In this paper, we first give some useful properties about k-dimensional numerical range of ξ -Lie product [ A , B ] ξ = A B - ξ B A for A , B ∈ B s (H) ; and then, based on these properties, all surjective maps preserving k-dimensional numerical ranges of ξ -Lie product on B s (H) are completely characterized. [ABSTRACT FROM AUTHOR]

Moreover, for some "mysterious" reasons, the new sequence HT ht turns out to be an integer sequence! Indeed, it follows directly from (4) that (5) can be rewritten without division as HT ht . If HT ht is a square root of -1, then I A i is a complex number, called a Gaussian integer. But if HT ht satisfies the condition HT ht then I A i is called a dual number. [Extracted from the article]

A vector generalization of the main concepts in the theory of functions of a complex variable— the concepts of the modulus and the argument of the complex number—is proposed. The authors introduce a certain generalization of the concept of holomorphic functions and mappings in the case of continuum complex spaces. [ABSTRACT FROM AUTHOR]

The problem for the Bessel equation of an integer order with complex physical and spectral parameters in the boundary condition is considered. The spectral parameter enters the boundary condition quadratically. The question of the basis property of the system of eigenfunctions in the case of the appearance of a multiple eigenvalue is studied. [ABSTRACT FROM AUTHOR]

This work presents a novel approach to quantum computing by proposing a customizable hardware design of a dedicated processor that emulates the execution of quantum algorithms. Unlike software-based quantum computation simulators, which run on standard general-purpose computers and suffer from reduced performance, this hardware design, which is based on classical concepts of bits, registers and memories, aims to leverage pure parallelism and pipelined execution for efficient quantum computations via emulation. The architecture includes several key components: memories, computation unit, measurement unit and control unit. The quantum state memory stores the individual and group states of qubits. This memory is crucial for maintaining the quantum information required for quantum operations. Basic operators are stored in dedicated operator memory. Additionally, a scratch memory allows for larger operators to be dynamically built at runtime. The computation unit is responsible for performing complex number multiplications, which form the basis of tensor and matrix products necessary for executing quantum operations. A measurement unit enables quantum state sampling, which is an essential aspect of quantum computation. Furthermore, a control unit is incorporated to ensure the correct operation of the quantum processor's data path. It utilizes a microprogram to manage and coordinate the functional units. All the functional units communicate with each other through dedicated and shared data buses, depending on the frequency of information exchange. This enables efficient data transfer and coordination among the components. The proposed hardware design has been simulated and proved to be effective in executing quantum operations. By exploiting parallelism and employing a pipelined execution, this architecture overcomes the limitations of software-based simulators, delivering improved performance for emulating quantum algorithms. We use Grover's search algorithm as a benchmark to evaluate the performance of the proposed hardware design and compare it to software-based simulation and to hardware-based algorithm-dedicated emulation. [ABSTRACT FROM AUTHOR]

A set S of points in R n is called a rationally parameterizable hypersurface if there is vector function σ : R n - 1 → R n having as components rational functions defined on some common domain D such that S = { σ (t) : t ∈ D } . A generalized n-dimensional polytope in R n is a union of a finite number of convex n-dimensional polytopes in R n . The Fourier–Laplace transform of such a generalized polytope P in R n is defined by F P (z) = ∫ P e z · x dx . Let γ be a fixed nonzero complex number. We prove that F P 1 (γ σ (t)) = F P 2 (γ σ (t)) for all t ∈ O implies P 1 = P 2 if O is an open subset of D satisfying some well-defined conditions and we present similar results for the null set of the Fourier–Laplace transform of P . Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres. [ABSTRACT FROM AUTHOR]

The concept of complex fuzzy set (CFS) and complex intuitionistic fuzzy set (CIFS) is two recent developments in the field of fuzzy set theory. The significance of these concepts lies in the fact that these concepts assigned membership grades from the unit circle in the plane, i.e., in the form of a complex number instead of [0,1] interval. CFS cannot deal with information of yes and no type, while CIFS works only for a limited range of values. To deal with these kinds of problems, in this article, the further development of the theory of complex Pythagorean fuzzy sets (CPFSs) has been discussed. Some new operations on CPFSs, such as bounded difference, disjoint sum, disjunctive sum, probabilistic sum, bold sum, bold intersection, s-norm and t-norm have been proposed. The distance of two CPFSs has been proposed. This distance measure is then used to define η -equalities of CPFSs. Moreover, we define some new notions for the multicriteria decision-making (MCDM) problems such as the CPF decision-making matrix, CPF max and min decision-making matrix, and the distance measure between CPF max and min decision-making matrices. An MCDM method has been developed on the proposed novel notions. A real-life example demonstrates that the MCDM method developed in the paper can be utilized to deal with problems of uncertainty. Further, the comparative study of CPFSs with complex intuitionistic fuzzy sets, Pythagorean fuzzy sets, intuitionistic fuzzy sets, complex fuzzy sets, and fuzzy sets is established. [ABSTRACT FROM AUTHOR]

The effect of an external electric field on the quark matter is an important question due to the presence of strong electric fields in heavy ion collisions. In the lattice QCD approach, the case of a real electric field suffers from the ‘sign problem’, and a classical electric field is often used similar as the case of chemical potential. Interestingly, in axial gauge a uniform classical electric field actually can correspond to an inhomogeneous imaginary chemical potential that varies with coordinate. On the other hand, with imaginary chemical potential, Roberge-Weiss (R-W) phase transition occurs. In this work, the case of a uniform classical electric field is studied by using lattice QCD approach, with the emphasis on the properties of the R-W phase. Novel phenomena show up at high temperatures. It is found that, the chiral condensation oscillates with z at high temperatures, and so is the absolute value of the Polyakov loop. It is verified that the charge density also oscillates with z at high temperatures. The Polyakov loop can be described by an ansatz Ap + Σq=u,dCq exp (LτQqiazeEz), where Ap is a complex number and Cd> 0, Cu ≥ 0 are real numbers that are fitted for different temperatures and electric field strengths. As a consequence, the behavior of the phase of Polyakov loop is different depending on whether the Polyakov loop encloses the origin, which implies a possible phase transition. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to understand Cowen–Douglas operators by the way of basis theory and shift operator. For a Cowen–Douglas operator T ∈ B n (Ω) and a complex number z 0 ∈ Ω , we show that there is a generalized basis { g k } k = 0 ∞ , such that the adjoint operator (T - z 0) ∗ is the forward shift on { g k } k = 0 ∞ , and if n ≥ 2 , then T - z 0 never is a backward shift on any Markushevich basis. Moreover, we give a characterization of a Cowen–Douglas operator in B 1 (Ω) being a backward shift on some Markushevich basis. Also, we show that a multiplication operator defined on Bergman space with a M o ¨ bius transformation as its multiplier is a forward shift on some Markushevich basis. [ABSTRACT FROM AUTHOR]

The dynamical stability of strange-dwarf hybrid stars and strange planets, constituted by strange-quark-matter cores and dilute-nuclear-matter crusts, is revisited by analyzing the fundamental mode eigenfrequencies of the radial oscillation equations with boundary conditions for slow (rapid) conversions originating at the density-discontinuous interface characterizing extremely large (small) microscopic timescales compared to the oscillation periods. For the hadronic crust we used an analytic fit of the BPS results matched to the massless MIT bag model. For the rapid case, our calculations indicate that the zero mode is the so-called reaction mode whose frequency is a complex number, thus ruling out the existence of strange dwarfs (planets) in nature. On the other hand, slow conversions still provide a sizeable stability window which, interestingly, also reproduces the Glendenning-Kettner-Weber results. The robustness of our findings is demonstrated for different transition densities and using an equation of state from perturbative QCD for the ultra-dense core. [ABSTRACT FROM AUTHOR]

In this paper, we study meromorphic solutions of the following nonlinear complex difference equation f n (z) + P (z) Δ η l f (z) = H 0 (z) + H 1 (z) e p 1 (z) + ⋯ + H m (z) e p m (z) , where n, l, m, q are positive integers with n ≥ 2 , η is a nonzero complex number satisfying Δ η l f (z) ≢ 0 , p 1 , ... , p m are polynomials of degree q, whose leading coefficients α 1 , ... , α m are distinct nonzero complex numbers, and P , H 0 , H 1 , ... , H m are meromorphic functions of order less than q such that P H 1 ... H m ≢ 0. In addition, we analyze meromorphic solutions of the above equation for n = 1 . Our proofs depend on Cartan's theorem and the variation of Nevanlinna's theorem regarding a set of meromorphic functions. Several examples are provided to demonstrate our results. These results generalize some very recent known results. [ABSTRACT FROM AUTHOR]

The authors consider the multidigit multiplication operation, on whose speed the speed of asymmetric cryptographic software and hardware depends. Algorithms for implementing the multiplication of two N-digit numbers based on discrete cosine and sine transforms (DCT and DST) are proposed. Due to the use of DCT and DST, the calculations for the real and imaginary parts of the discrete Fourier transform (DFT) of a real even-length signal are separated, which allows translating the complex-number calculations to real-number calculations. The operations of the algorithm are replaced in order to preserve symmetry in the real or imaginary parts of the multidigit numbers, which allows the use of DCT and DST of lower length N / 2 + 1 and increases the possibility of parallelizing in multidigit multiplication. [ABSTRACT FROM AUTHOR]

The q-rational numbers and the q-irrational numbers were introduced by Morier-Genoud and Ovsienko. In this paper, we focus on q-real quadratic irrational numbers, especially q-metallic numbers and q-rational sequences which converge to q-metallic numbers, and consider the radiuses of convergence of them when we assume that q is a complex number. We construct two sequences given by recurrence formula as a generalization of the q-deformation of the Fibonacci and Pell numbers which are introduced by Morier-Genoud and Ovsienko. For these two sequences, we prove a conjecture of Leclere, Morier-Genoud, Ovsienko and Veselov concerning the expected lower bound of the radiuses of convergence. In addition, we obtain a relationship between the radius of convergence of these two sequences in two special cases. [ABSTRACT FROM AUTHOR]

We establish some -inequalities for the polar derivative of a polynomial. Our results generalize some known Bernstein-type polynomial inequalities and include as special cases several interesting generalizations of some Zygmund-type inequalities for polynomials as well. [ABSTRACT FROM AUTHOR]

Since the eigenfrequency of gravitational waves from cold neutron stars becomes a complex number, where the real and imaginary parts respectively correspond to an oscillation frequency and damping rate, one has to somehow solve the eigenvalue problem concerning the eigenvalue in two-dimensional parameter space. To avoid this bother, one sometimes adopts an approximation, where the eigenvalue is in one-dimensional parameter space. In this study, first, we show the accuracy of the zero-damping approximation, which is one of the one-dimensional approximations, for the fundamental and 1st pressure modes. But, this approximation is not applicable to the spacetime mode, because the damping rate of the spacetime mode is generally comparable to the oscillation frequency. Nevertheless, we find the empirical relation for the ratio of the imaginary part to the real part of the eigenfrequency, which is expressed as a function of the steller compactness almost independently of the adopted equations of state for neutron star matter. Adopting this empirical relation, one can express the eigenfrequency in terms of just the real part, i.e., the problem to solve becomes an eigenvalue problem with a one-dimensional eigenvalue. Then, we find that the frequencies are estimated with good accuracy even with such approximations even for the 1st spacetime mode. [ABSTRACT FROM AUTHOR]

We prove that any complex values z 1 , ... , z n can be approximated by the following integral shifts of the Riemann zeta-function ζ (s + i d 1 τ) , ... , ζ (s + i d n τ) for infinitely many τ , provided d 1 , ... , d n ∈ Q are distinct and positive, and s is a fixed complex number lying in the right open half of the critical strip. [ABSTRACT FROM AUTHOR]

The central limit theorem has been found to apply to random vectors in complex Hilbert space. This amounts to sufficient reason to study the complex–valued Gaussian, looking for relevance to quantum mechanics. Here we show that the Gaussian, with all terms fully complex, acting as a propagator, leads to Schrödinger’s non-relativistic equation including scalar and vector potentials, assuming only that the norm is conserved. No physical laws need to be postulated a priori. It thereby presents as a process of irregular motion analogous to the real random walk but executed under the rules of the complex number system. There is a standard view that Schrödinger’s equation is deterministic, whereas wavefunction “collapse” is probabilistic (by Born’s rule)—we have now a demonstrated linkage to the central limit theorem, indicating a stochastic picture at the foundation of Schrödinger’s equation itself. It may be an example of Wheeler’s “It from bit” with “No underlying law”. Reasons for the primary role of C are open to discussion. The present derivation is compared with recent reconstructions of the quantum formalism, which have the aim of rationalizing its obscurities. [ABSTRACT FROM AUTHOR]

Fast Fourier Transform is widely used in communication and signal processing. I propose an improved multipath delay commutator pipelining architecture based on the radix-2 time decimation algorithm. By optimizing the intermediate data processing process and the first stage of pipelining, the architecture improves the system's computing speed and reduces the use of registers. I propose a multiplication scheme based on CORDIC and binary decomposition coding to realize complex number multiplication and constant multiplication and to eliminate the use of a multiplier. Experimental results suggest that proposed implementation has less latency and hardware utilization as compared to recently proposed implementations. [ABSTRACT FROM AUTHOR]

An extensive variety of human decision making is based on two-sided or bipolar cognitive assessment of a positive side and a negative side. Zhang's bipolar fuzzy set is represented as { ∀ (x , y) | (x , y) ∈ [ - 1 , 0 ] × [ 0 , 1 ] }. The multiagent decision analysis is based on bipolar judgment of a positive side and a negative side. In this paper, we consider the issues posed by bipolar fuzzy linear systems (BFLSs) and bipolar fuzzy complex linear (BFCL) systems of equations with real and complex coefficients. Primarily, we find the solution of the BFLS of equations by the resort into two parts: one is negative (∙) and other is positive (⋆) n × n linear fuzzy systems; to this aim, we use n × n bipolar fuzzy center system and 2 n × 2 n bipolar fuzzy width system. We describe the fundamentals of proposed method with the aid of n = 2 , 5 -dimensional numerical examples, and we get the weak and strong solutions of the system. Subsequently, we describe a technique to solve BFCL systems of equations whose coefficients are real and complex by a pair of positive (⋆) and negative (∙) n × n real and complex crisp linear systems; to this aim, we use bipolar fuzzy complex center and width. Ultimately, we use our proposed technique to solve a current flow circuit which is defined in terms of a BFCL system with complex coefficients, and we get the unknown current in the form of bipolar fuzzy complex number. [ABSTRACT FROM AUTHOR]

Given a complex number a, we prove that there is at most one normalized Hecke eigenform of level one whose third Fourier coefficient is given by a, assuming that the characteristic polynomials of the Hecke operators T 3 are irreducible. This generalizes a result of Vilardi and Xue on the second Fourier coefficients. Under the assumption on irreducibility of characteristic polynomials of T 2 and T 3 , we further show that there is at most one normalized eigenform of level one whose second or third Fourier coefficient is given by a. [ABSTRACT FROM AUTHOR]

We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any α -fold divisor function, for any complex number α ∉ { 1 } ∪ - N , even when considering a sequence of parameters α close in a proper way to 1. Our work builds on that of Harper and Soundararajan, who handled the particular case of k-fold divisor functions d k (n) , with k ∈ N ≥ 2 . [ABSTRACT FROM AUTHOR]

Background: In the existing literature, the dynamic characteristics of the spinning shaft-disk systems are usually evaluated using the transfer matrix method (TMM), the finite element method (FEM) or the differential quadrature method (DQM). Purpose: The results of the above-mentioned TMM, FEM or DQM are the approximate solutions and the exact solution concerned is rare. For this reason, this paper aims at presenting an analytical method to yield the exact whirling speeds and mode shapes of a "multi-span" continuous shaft mounted by arbitrary rigid disks. Method: In theory, the whirling motion of a rotating shaft-disk system is three-dimensional (3-D), however, if the transverse displacement in the vertical principal xy-plane and that in the horizontal principal xz-plane for the cross-section of the rotating shaft located at axial coordinate x are combined using a complex number, and the effects of each rigid disk i mounted on the shaft are replaced by a lumped mass together with a frequency-dependent "equivalent" mass moment of inertia, then the whirling motion of a rotating shaft-disk system will be similar to the two-dimensional (2-D) transverse free vibration of a stationary beam. After the foregoing manipulations, the techniques for the free vibrations of a stationary beam carrying various concentrated elements (including the intermediate supports) will be available for the whirling motions of a rotating "multi-span" shaft mounted by arbitrary rigid disks. Results and Conclusions: To confirm the reliability of presented theory and developed computer programs, most results obtained from the presented method are compared with the available literature or the results of FEM and good agreements are achieved. The presented method has the following merits: (a) The complicated 3-D whirling motions of a rotating shaft can be solved with the approaches for the simple 2-D free vibrations of a stationary beam, so that much computer time can be saved. (b) The damping matrix is eliminated so that the computer programming is easier. (c) The obtained results are the "exact" solutions and may be the benchmark for evaluating the accuracy of the other approximate methods. [ABSTRACT FROM AUTHOR]

Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces1,2. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural3. In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states4. Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.A Bell-like experiment that discriminates between real-number and complex-number multipartite quantum systems could disprove real quantum theory. [ABSTRACT FROM AUTHOR]

We consider polynomials of the form and prove some results for the estimate of the polar derivative and generalize the results due to Aziz and Shah [4], Govil [12] and others. [ABSTRACT FROM AUTHOR]